Restricted frame graphs and a conjecture of Scott

TL;DR

This paper investigates the limitations of Scott's conjecture on graph coloring, providing new counterexamples by analyzing specific graph constructions, especially those derived from subdividing edges of certain multigraphs.

Contribution

It extends previous counterexamples to Scott's conjecture by identifying new classes of graphs obtained through edge subdivisions that disprove the conjecture.

Findings

Scott's conjecture is false for graphs from subdividing edges of K4.

Counterexamples exist for graphs derived from 2-connected multigraphs with no universal cycle vertex.

The study deepens understanding of which graphs violate Scott's conjecture.

Abstract

Scott proved in 1997 that for any tree $T$, every graph with bounded clique number which does not contain any subdivision of $T$ as an induced subgraph has bounded chromatic number. Scott also conjectured that the same should hold if $T$ is replaced by any graph $H$. Pawlik et al. recently constructed a family of triangle-free intersection graphs of segments in the plane with unbounded chromatic number (thereby disproving an old conjecture of Erd\H{o}s). This shows that Scott's conjecture is false whenever $H$ is obtained from a non-planar graph by subdividing every edge at least once. It remains interesting to decide which graphs $H$ satisfy Scott's conjecture and which do not. In this paper, we study the construction of Pawlik et al. in more details to extract more counterexamples to Scott's conjecture. For example, we show that Scott's conjecture is false for any graph obtained from $K_4$ by subdividing every edge at least once. We also prove that if $G$ is a 2-connected multigraph with no vertex contained in every cycle of $G$, then any graph obtained from $G$ by subdividing every edge at least twice is a counterexample to Scott's conjecture.